Simplify the following expression and state the condition under which the simplification is valid: $x = \dfrac{n^2 - 2n - 3}{n^2 - 6n - 7}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{n^2 - 2n - 3}{n^2 - 6n - 7} = \dfrac{(n - 3)(n + 1)}{(n - 7)(n + 1)} $ Notice that the term $(n + 1)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(n + 1)$ gives: $x = \dfrac{n - 3}{n - 7}$ Since we divided by $(n + 1)$, $n \neq -1$. $x = \dfrac{n - 3}{n - 7}; \space n \neq -1$